Dark bubbles: decorating the wall
UUUITP-1/20
Dark bubbles: decorating the wall
Souvik Banerjee, a Ulf Danielsson, b Suvendu Giri b a Institut f¨ur Theoretische Physik und Astrophysik, Julius-Maximilians-Universit¨at W¨urzburg,AmHubland, 97074 W¨urzburg, Germany b Institutionen f¨or fysik och astronomi, Uppsala Universitet, Box 803, SE-751 08 Uppsala, Sweden
E-mail: [email protected] , [email protected] , [email protected] Abstract:
Motivated by the difficulty of constructing de Sitter vacua in string theory, a newapproach was proposed in arXiv:1807.01570 and arXiv:1907.04268, where four dimensionalde Sitter space was realized as the effective cosmology, with matter and radiation, on anexpanding spherical bubble that mediates the decay of non supersymmetric AdS to a morestable AdS in string theory. In this third installment, we further expand on this scenarioby considering the backreaction of matter in the bulk and on the brane in terms of howthe brane bends. We compute the back reacted metric on the bent brane as well as in thefive dimensional bulk. To further illuminate the effect of brane-bending, we compare ourresults with an explicit computation of the five dimensional graviton propagator using aholographic prescription. Finally we comment on a possible localization of four dimensionalgravity in our model using two colliding branes. a r X i v : . [ h e p - t h ] A p r ontents A Fourier transform of K The extreme difficulty of constructing a model of the observed positive cosmological constantin string theory, has led to the recent proposal of de Sitter (dS) swampland conjectures[1, 2, 3], which forbid the existence of such vacua in string theory (see [4] for a review).To get around this difficulty, a possible new approach was proposed in [5, 6], where fourdimensional dS is realized as a time-dependent geometry in an unstable five dimensionalAdS vacuum of string theory. This scenario is inspired by the Randall-Sundrum braneworld[7, 8] construction, but differs from it in a crucial way. Instead of a flat brane with a Z symmetry across it, our shellworld is a spherical brane that mediates the decay of anunstable five dimensional AdS to a more stable one. This leads to important differencesin the way that four dimensional matter and radiation is realized on the shellworld. Forinstance, a cloud of strings stretching out from the shell leads to matter, while a black holein the bulk give rise to radiation on the four dimensional shellworld. In fact, it was shown in[9] that the presence of these ingredients in the five dimensional bulk facilitate the decay ofthe unstable AdS and favor the formation of a bubble with a small cosmological constant.Let us begin by summarizing the shellworld construction of [5, 6]. Consider a theorythat has two AdS vacua with cosmological constants Λ + (= − k = − /L ) and Λ − (= − k − = − /L − ) with k − > k + . The vacuum with higher energy Λ + can decay into thevacuum with lower energy Λ − non-perturbatively via the nucleation of a Brown-Teitelboiminstanton. The decay AdS vacua supported by flux through the nucleation of charged In string theory, such decays are supported by the conjecture that all non supersymmetric AdS vacuamust decay as a consequence of the weak gravity conjecture [10, 11, 12]. – 1 –embranes, and its relation to the Weak Gravity Conjecture [13] in the context of shellworlds,was discussed in [6]. Let us consider these AdS vacua in global coordinates (where thesubscripts + and − denote quantities outside and inside the bubble respectively)d s = − f ± ( r )d t + f ± ( r ) − d r + r dΩ , (1.1)where f ± ( r ) = 1 + k ± r . For the metric inside the bubble with f − ( r ), the radial coordinate r goes from the centre of AdS to the position of the bubble i.e. r ∈ (0 , a ( t )), while forthe metric outside the bubble with f + ( r ), r ∈ ( a ( t ) , ∞ ). The complete metric for all r i.e. r ∈ (0 , ∞ ) in the five dimensional bulk is then given byd s = − f ( r )d t + f ( r ) − d r + r dΩ , (1.2)with f ( r ) := (1 + k − r ) + Θ( r − a ( t ))( k r − k − r ) , (1.3)On parametrizing the radius of the bubble in terms of the proper time ( τ ) for anobserver on the shell i.e. , r = a ( τ ), the induced metric takes the FLRW formd s = − d τ + a ( τ ) dΩ . (1.4)Einstein’s equations require the presence of a stress tensor on the shell given by S ab = − κ (cid:0) [ K ab ] + − − [ K ] + − γ ab (cid:1) , (1.5)where κ ≡ πG , and γ ab is the metric induced on the shell from the bulk, where theindices ( a, b ) ∈ { , , , } run over the shell. K ab is the extrinsic curvature of the shellas seen from the AdS bulk, and [ · ] + − := ( · ) + − ( · ) − is the difference of the correspondingquantity between the outside and the inside of the bubble. This governs the evolution ofthe radius of the bubble, and for a shell of constant tension σ it is given by σ = 3 κ (cid:32)(cid:114) k − + 1 + ˙ a a − (cid:114) k + 1 + ˙ a a (cid:33) , (1.6)where a ( τ ) ≡ a and ˙ a := d a ( τ ) / d τ . For a critical value of the tension σ crit given by σ crit := 3 κ ( k − − k + ) , (1.7)the cosmological constant on the shell vanishes, giving a four dimensional Minkowski brane.For a brane with a slightly sub-critical tension σ = σ crit (1 − (cid:15) ), equation (1.6), up to linearorder in (cid:15) , gives the Friedmann equation H ≡ ˙ a a = − a + κ + O (cid:0) (cid:15) (cid:1) , (1.8) Inside (outside) refers to the direction away from the brane in which the volume of radial slices decreases(increases). We follow the standard conventions where the coefficient of the five dimensional Ricci scalar in theEinstein Hilbert action is M = 1 / (2 κ ) = 1 / (16 πG ), where M is the five dimensional Planck mass. – 2 –here κ ≡ πG ≡ k + k − k − − k + κ , Λ ≡ σ crit − σ = (cid:15)σ crit . (1.9)Therefore, an observer living on such an expanding shell experiences a de Sitter universewith a positive spatial curvature.In the presence of matter in the five dimensional AdS, the metric is given by equation(1.1) with f ± ( r ) = 1 + k ± − κ M ± / (3 π r ). This gives an additional contribution to theFriedmann equation which goes as 1 /a and can be identified as a radiation density in fourdimensions i.e. , H = − a + κ (cid:20) Λ + 12 π a (cid:18) M + k + − M − k − (cid:19)(cid:21) . (1.10)Four dimensional matter can be obtained from a cloud of strings in the bulk, which has f ± ( r ) = 1 + k ± − κ M ± / (3 π r ) − κ α/ (4 πr ), and gives the following contribution to theFriedmann equation H = − a + κ (cid:20) Λ + 12 π a (cid:18) M + k + − M − k − (cid:19) + 38 πa (cid:18) α + k + − α − k − (cid:19)(cid:21) . (1.11)In summary, a positively spatially curved four dimensional de Sitter universe can be modeledas the effective spacetime seen by a four dimensional observer living on the surface of anexpanding shell in an asymptotically five dimensional AdS. The matter and radiationdensities observed in the four dimensional universe are induced by the presence of matterand a cloud of strings in the five dimensional bulk.This model also provides a way of understanding the dS horizon in four dimensions[6]. A four dimensional observer living on the shell accelerates in the fifth dimension withthe expanding shell and measures an Unruh temperature. This turns out to be equal tothe temperature associated with the dS horizon in four dimensions. The dS temperaturethus has a five dimensional interpretation in terms of the acceleration of the expandingshell. This means that the cosmological horizon is a section of the Rindler horizon in fivedimensions.Gravitational dynamics on the shell can be studied using the Gauss (-Codazzi) equations,which relate the Einstein equations in the bulk to those on the shell in terms of its embeddingand extrinsic curvature, R (5) αβγδ e αc e βa e γd e δb = R (4) cadb + ( K ad K cb − K cd K ab ) . (1.12)Combined with the thin shell junction conditions, it was shown in [6] that the four dimen-sional Einstein equations are given by (cid:16) G (4) (cid:17) ab = − k + k − (cid:18) − κ k − − k + σ (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) ≡ κ ( σ crit − σ ) ≡ κ Λ δ ab − κ π a ( τ ) (cid:18) M + k − − M − k + k − − k + (cid:19) (cid:32) δ a δ b − (cid:88) i =1 δ ai δ ib (cid:33) − κ πa ( τ ) (cid:18) α + k − − α − k + k − − k + (cid:19) δ a δ b , (1.13)– 3 –he left hand side of this equation is the Einstein tensor, and the right hand side gives thesources corresponding to the four dimensional cosmological constant, radiation and matterrespectively. Each contribution is easily identified through its equation of state given by p = − ρ, p = ρ/ p = 0, respectively. Taking the ( τ, τ ) component of the Einstein tensorin four dimensions, namely G (4) ττ = 3 (cid:0) /a + ˙ a /a (cid:1) and using the above equation, givesthe Friedmann equation (1.11) as expected.In the current paper, we will re-derive (1.9) through a computation of the gravitonpropagator in momentum space. In this computation, we will explicitly use intuitionscoming from holography [14]. Holography, in particular the AdS/CFT correspondence[15], relates a gravitational theory in ( d + 1) dimensional anti de Sitter spacetime to anon-gravitational d -dimensional conformal field theory living on the boundary of the ( d + 1)dimensional bulk spacetime. In its full generality, both the bulk and boundary theories arequantum theories. However, in certain limits the gravitational theory becomes classicalas well as weakly coupled. In this limit, there is a well-defined prescription [16, 17] thatrelates bulk quantities to boundary data. In particular, the bulk modes that decay withradial distance near the boundary (termed as the normalizable modes ) are interpreted asthe expectation value of the corresponding operator on the quantum field theory on theboundary, while the other independent set of relatively slowly growing or constant modes(namely, the non-normalizable modes ) have the interpretation of sources. As an example,for the graviton propagator, normalizable modes correspond to the stress tensor while thenon-normalizable modes correspond to the metric on the boundary. We will not get intomuch of the technical details of holographic prescription in this paper , but only use thisintuition while fixing the constants of integration when computing the bulk propagator.Our main focus in this paper will be the following. In the construction of [5, 6], theshell is radially symmetric and is located at a constant radius in the five dimensional bulkat each instant of time. The presence of four dimensional Einstein gravity on the brane wasshown in the presence of homogeneous and isotropic matter. However, in the presence oflocalized matter on the brane, breaking homogeneity and isotropy, it is expected to bend[19, 20]. This backreaction, which has not been considered so far, is one of the main aspectsthat we will focus on in this paper. In addition, we will also compute the metric on thebent brane as well as the back-reacted five dimensional metric.In figure 1, we summarize the contribution of matter sources to the bending of the brane.We will discuss this in detail in section 2 and here we present a geometrical understandingof the main result. In the coordinates that we are using, the Poincar´e horizon is located at ξ → −∞ and the boundary of AdS is at ξ → ∞ , so gravity pulls down towards the centerof the AdS throat, implying that matter confined to the brane will make it sag (case (a) infigure 1). The brane will bend in a similar way with a string pulling downwards (case (b)in figure 1). As we will argue in subsequent sections, it is the case with strings pulling thebrane upwards (case (c) in figure 1) that will be the most interesting. The presence of suchstrings will affect the metric in the bulk and on the brane in two different ways. There is namely, in the limit of large number of degrees of freedom, N → ∞ in the boundary field theory andlarge t’Hooft coupling λ = g qft N (cid:29) g qft being the coupling in the boundary field theory. Interested readers are referred to the elaborate review [18] and references therein. – 4 – = 0 x a ξ (a) (b) (c) k − k + Figure 1 : A schematic representation of the brane, showing the effect of various forms ofstress tensor. The brane is shown in black and the sources are in red. (a) A point masscauses the brane to “sag” i.e. , ξ < . A gauge transformation to bring the brane back upto ξ = 0 is physically equivalent to adding a negative four dimensional mass to the braneto “float” it back up. (b) A string pulling on the brane from the inside has the same effect,while (c) a string pulling from outside has the opposite effect and contributes with a positivefour dimensional energy in a gauge where the brane is flat. the change in the metric sourced directly by the string that is present even without thebrane, as well as a contribution from the bending through the junction conditions. As wewill see explicitly in section 3.2., the direct effect is subleading at distances much largerthan the AdS-scale, where the effect from the bending dominates.In order to distinguish between the two effects, let us invoke the covariant conservationof the stress tensor i.e. , ∇ α T αβ = 0. For β = ξ , the conservation condition gives (assumingno cross terms in ξ i.e. , T aξ = 0) ∂ ξ T ξξ + 4 kT ξξ + ka ( ξ ) η ab T ab = 0 . (1.14)Recalling that the metric is d s = d ξ + a ( ξ ) η ab d x a d x b , a string of constant tension τ stretching only in the ξ direction (and located at x a = x a ) has an energy density given by T = T ξξ = τ a ( ξ ) δ ( x a − x a ) . (1.15)This corresponds to an equation of state p = ρ along the fifth direction and p = 0 in thetransverse direction, which is exactly as expected for a string stretching out infinitely only The Bianchi identities require the stress tensor to be covariantly conserved in order for Einstein’sequations to be consistent. This is a constraint on the components of the stress tensor that we will use hereto find the stress tensor of the end point of the string in terms of the tension of the string. This is similar tothe treatment in [20]. – 5 –n the ξ direction. If, instead, the string ends on the brane (located at ξ = ξ , say), then T ξξ becomes T ξξ = τ a ( ξ ) δ ( x a − x a ) Θ ( ξ − ξ ) . (1.16)Equation (1.14) then gives T = τ a ( ξ ) δ ( x a − x a ) Θ ( ξ − ξ ) (cid:124) (cid:123)(cid:122) (cid:125) string stretching in the bulk + τk a ( ξ ) δ ( x a − x a ) δ ( ξ − ξ ) (cid:124) (cid:123)(cid:122) (cid:125) end point on the brane . (1.17)The first term is the expected energy density of the string stretching in the bulk, but nowthere is a second contribution on the brane corresponding to the point where the stringends on the brane. This is the energy density of a point particle with mass τ /k exactly aswas argued from the Friedmann equations in [6]. It is useful to think of this as a point masshanging by a string in the gravitational well of the AdS-space. As the universe expands,and the brane moves upwards, the string pulls the point mass so that it can move alongwith the brane.This is the setup that we will be revisiting throughout this paper. The organizationis as follows. In section 2, we will introduce the idea of brane bending in the presence ofmatter sources. We will start with a leading order computation demonstrating the effect ofbrane bending and then perform a more detailed analysis to show which way the branebends in response to being pulled by strings in the bulk. In section 3, we will compute thegraviton propagator first in momentum space and then in position space. Subsequently, wewill discuss how these computations are related by a gauge choice. Finally, in section 4, wewill highlight some future directions. In particular, we will discuss the possibility of fullylocalizing gravity in the shellworld model using two approaching bubbles, the possibility ofrealizing black holes on the shellworld and a realization of holography on a cut-off brane. In this section, we will study the effect of brane bending in response to matter (both in thebulk and on the brane) that we briefly introduced in section 1. It will be convenient towork in coordinates where the AdS metric has conformally flat slices along a chosen fifthdirection ξ i.e. , AdS ≡ R × a ( ξ ) MkW .d s = d ξ + a ( ξ ) η ab d x a d x b , (2.1)where a ( ξ ) := exp( kξ ), 1 /k = L is the AdS radius, indices ( a, b ) ∈ { , , , } as before, η ab is the Minkowski metric in four dimensions and Greek letter indices are five dimensional i.e. , x α ∈ ( ξ, x a ).Let us choose two AdS spaces with radii L ± = 1 /k ± in the regions ξ ≷ ξ ∈ R to be a solution of Einstein’s equations,a source needs to be present at the discontinuity at ξ = 0. In this case, the source isa brane situated at ξ = 0 which has a stress tensor proportional to the difference of itsextrinsic curvatures as seen from either side. This, along with the uniqueness of the induced– 6 –etric on the brane goes by the name of Israel-Lanczos-Sen’s thin shell junction conditions [21, 22, 23], [ γ ab ] + − = 0 ,S ab = − κ (cid:0) [ K ab ] + − − [ K ] + − γ ab (cid:1) , (2.2)where the extrinsic curvature K ab is defined (in terms of the normal to the shell n α andtangents e αa := d x α / d y a ) as K ab = n α ; β e αa e βb . In these coordinates, the extrinsic curvaturebecomes K ab = (1 / ∂ ξ γ ab . Note that the second condition is the same as the one we usedin (1.5) while the first condition was automatic in section 1 when choosing proper time onthe shell.In the absence of matter on the brane and in the limit of large proper radius ( ar → ∞ ),the brane can be taken to be flat and lie at ξ = 0. However, in the presence of matter, thebrane is not expected to be flat any more. The deformation of the brane in the presence ofmatter was discussed in [19, 20] and is usually referred to as brane bending . Before analyzingthe effect of bending in detail for our setup, let us first look at a matter perturbation andthe corresponding bending at leading order.Let us consider the perturbation of the bulk metric in the presence of a small amount ofmatter on the brane (resulting from the presence of strings in the bulk). Off the brane andaway from the strings, there are no sources and the metric can be chosen to be transverseand traceless and is given by the junction conditions. There are two perturbation parametersin this case – first, the matter density ( χ ) and second, the inverse of the proper radius ( ar ).The metric can be thought of as a double expansion in these parameters χ/r and 1 / ( kar ).Up to leading order in large proper radius, a transverse traceless perturbation in responseto matter density χ is given by d s = d ξ + a ( ξ ) (cid:20) − (cid:18) − χr (cid:19) d t + (cid:18) χr (cid:19) d r + (cid:16) χr (cid:17) r dΩ (cid:21) . (2.3)The position of the brane taking into account the bending is given by ξ = ˜ ξ + f ( r ). Shiftingthe radial coordinate by a constant r = ˜ r − χ and dropping quadratic terms in f or itsderivatives givesd s = d ˜ ξ + a ( ˜ ξ ) (cid:20) − (cid:18) − χr + 2 kf ( r ) (cid:19) d t + (cid:18) χr + 2 kf ( r ) (cid:19) d r + r dΩ (cid:21) , (2.4)For this to be a vacuum solution of Einstein’s equations at lowest order in χ/r and 1 / ( kar ), kf ( r ) = χ r = M r , (2.5)where χ = 2 M/
3, and M is the effective mass from the four dimensional point of view.Let us now do a more careful analysis, taking into account the presence of sources.Starting from equation (2.2) and splitting the stress tensor on the brane into the brane We will derive the metric to linear order in χ/r and all orders in ( kar ) − in section 3.2 and this metricis just the large proper radius expansion of the metric in equation (3.33) i.e. , expanded to ( kar ) . – 7 –ension σ (equation of state ρ = − p ) and other matter T ab i.e. , S ab = − σγ ab + T ab , thesecond junction condition becomes12 κ [ ∂ ξ γ ab ] + − + σ γ ab = − (cid:18) T ab − T γ ab (cid:19) . (2.6)In the absence of matter on the brane i.e. , T ab = 0, the brane is intrinsically flat and has acritical tension given by σ = 3 κ ( k − − k + ) . (2.7)Now consider a perturbation of the AdS metric due to the presence of some matter bothon the brane and in the bulk. These perturbations can be chosen to be Gauss normal i.e. , g αβ (cid:55)→ g αβ + δg αβ such that δg µξ = δg ξξ = 0 . (2.8)This causes a change in the induced metric γ ab (cid:55)→ γ ab + ˆ h ab , which in our coordinatesbecomes a ( ξ ) η ab (cid:55)→ a ( ξ ) η ab + ˆ h ab . The second junction condition then becomes (recallthat this is evaluated on the shell)12 κ (cid:104) ∂ ξ ˆ h ab (cid:105) + − + σ h ab = − (cid:18) T ab − T η ab (cid:19) , (2.9)where σ is the critical tension in equation (2.7).In the presence of matter, the no longer expected to be at ξ = 0. This makes it difficultto apply the junction conditions since the position of the brane would be given by twodifferent functions ξ = f + ( x ) and ξ = f − ( x ) as seen from either side of the brane. To getaround this, we can make a gauge transformation on the outside and another on the insideof the brane, following [19, 20]. Demanding that the new coordinates are also Gauss normaland that the brane is located at ξ = 0 in the new coordinates, we get, following [24], ξ (cid:55)→ ξ − f ( x ) , x a (cid:55)→ x a + 12 k (cid:0) − a − (cid:1) ∂ a f ( x ) + q a ( x ) , (2.10)where q a ( x ) is an arbitrary function of the transverse coordinates. As mentioned above, weneed to apply different coordinate transformations on either side of the brane to flatten itand so q a ( x ) , f ( x ) and k are different on either side of the brane. In these new coordinates,the the metric perturbation becomesˆ h ab = h ab + 1 k (cid:0) − a (cid:1) ∂ a ∂ b f + 2 kf γ ab − a ( ∂ a q b + ∂ b q a ) , (2.11)which when evaluated on the brane ( ξ = 0 ⇒ a = 1) becomes δγ ab = h ab + 2 kf η ab − q [ a,b ] , (2.12)where q [ a,b ] := (1 /
2) ( ∂ a q b + ∂ b q a ) is the anti-commutator. Continuity of the induced metric( δh ab = 0) requires [ h ab ] + − = [ kf ] + − = (cid:2) q [ a,b ] (cid:3) + − = 0 . (2.13)– 8 –t is practical to introduce the function F := kf since it is continuous across the shell. Usingthis, equation (2.9) becomes 12 κ [ ∂ ξ h ab ] + − + σ h ab = − Σ ab , (2.14)where Σ ab is defined asΣ ab = (cid:18) T ab − T η ab (cid:19) − κ (cid:18) k + − k − (cid:19) ∂ a ∂ b F, (2.15)which gives the contribution of brane bending on the stress tensor. Therefore, choosinga bent gauge (or equivalently, considering a bent brane) modifies the stress tensor on thebrane with a gauge dependent factor F = kf . This can be chosen so that Σ ab is traceless.This determines F in terms of the trace of the stress tensor on the brane T and explicitlyshows how the brane bends. This is analogous to the result in [24]. T = − κ (cid:18) k + − k − (cid:19) (cid:3) F. (2.16)Note that the factor 3 nicely cancels the 1 / kf ( r ) that we found in equation (2.5) bymatching against Schwarzschild. A point mass on the brane i.e. , T ∼ δ ( r ) > ⇒ T = − T ∼ − δ ( r ), gives (cid:3) F ∼ δ ( r ) ⇒ F ∼ − /r < ⇒ f <
0. In equation (2.10), we havedefined f ( r ) as the amount that the brane actually bends and so the coordinate ξ has to bepulled down by the same amount to get a flat brane. Placing a point mass on the brane( T ∼ δ ( r ) >
0) implies f ( r ) < i.e. , f ( r ) > In this section, we will first present key features of graviton propagators on our shell-world,starting from a bulk computation in momentum space. We will then repeat this computationin the position space, but now allowing for the brane to bend. This will in turn allow thepossibility of having massive structures on the shell as argued before. We will finally discusshow these two apparently different computations are related via a gauge choice.
Let us consider perturbations of the AdS metric in response to sources, both in the bulkand on the brane of the formd s = d ξ + a ( ξ ) ( η ab + h ab ( ξ, x a )) d x a d x b , (3.1)where the perturbations are taken to be Gauss normal. Here a ( ξ ) := exp( kξ ), 1 /k beingthe AdS length scale. – 9 –aking the transverse coordinates, x a , as standard Cartesian coordinates, the linearizedEinstein equations in the world volume directions of the brane are given by [20] (cid:3) ¯ h ab = a − (cid:16) − η ab ∂ c ∂ d ¯ h cd + ∂ c ∂ a ¯ h bc + ∂ c ∂ b ¯ h ac (cid:17) − η ab a − ∂ ξ (cid:0) a ∂ ξ ¯ h (cid:1) − κ a − T ab , (3.2)where a is a shorthand for a ( ξ ) , ¯ h ab is the trace reversed perturbation defined as ¯ h ab := h ab − η ab h, and h := h ab η ab is the trace with respect to the unperturbed metric. Outsidethe source, a transverse gauge can be chosen [20] i.e. , ∂ a ¯ h ab = 0, which implies Lorenzgauge for the metric perturbations h ab i.e. , ∂ a h ab = ∂ b h .It is easier to work with a perturbation which includes the metric factor a ( ξ ) ( i.e. , γ ab := a ( ξ ) h ab ) and its trace-reversed form ¯ γ ab = γ ab − η ab γ . Rewriting (3.2) in terms of¯ γ ab and taking a trace gives a − ∂ c ¯ γ + 3 (cid:0) ∂ ξ − k (cid:1) ¯ γ = − κ T, (3.3)where traces are taken with respect to the background metric i.e. , ¯ γ := ¯ γ ab η ab and T := T ab η ab . Subtracting this from (3.2) gives an equation for the traceless piece a − ∂ c χ ab + (cid:0) ∂ ξ − k (cid:1) χ ab = − κ Σ ab , (3.4)where χ ab and Σ ab are the traceless pieces of ¯ γ ab and T ab respectively i.e. ,¯ γ ab = χ ab + 14 η ab ¯ γ,T ab = Σ ab + 14 η ab T. (3.5)We will now try to solve for the traceless perturbation χ ab . This is easier to do in momentumspace. After a Fourier transformation, equation (3.4) becomes (cid:18) − p a + ∂ ξ − k (cid:19) ˜ χ ab ( p, ξ ) = − κ ˜Σ ab , (3.6)where the tilde represents the Fourier transform in the transverse directions. p = −| p | + | (cid:126)p | is negative for pure AdS spacetime. p can be positive for spacelike modes with | p | < | (cid:126)p | . Such modes can appear, for example, when there is a black hole in the bulkspacetime. In our case, with the source on the shell, we no longer have pure AdS spacetime.Let us use this fact to postulate that such non-propagating modes do exist in our case, andlet us study the gravitational propagator between two bulk points across the shell for thesemodes. We will justify our claim through a position space computation in the next section.Equation (3.6) suggests that the structure of propagator for the modes, ˜ χ ab should bevery similar to that of a minimally coupled massless scalar propagator in this spacetime. In Holographically, these modes are commonly interpreted as thermal modes in the boundary quantumfield theory. Note, in zero temperature field theory, we can never have such modes because of spectrumcondition ( ω > k ). However, when we connect the system to a heat reservoir, particles with arbitrarily largemomenta can come out of the reservoir or go in keeping the total energy of the system lie within a narrowband. – 10 –rder to see this similarity explicitly, let us consider a minimally coupled massless scalarfield in AdS and the Green function for the scalar Laplacian in the bulk i.e. , (cid:3) ∆ (cid:16) X α , ˜ X α (cid:17) = δ ( X α − ˜ X α ) √− G , (3.7)where, as before, X α ≡ { ξ, x c } . α runs from 0 to 4, c runs from 0 to 3 and ξ is the fifthdirection. In the background of pure AdS spacetime,d s = d ξ + a ( ξ ) η ab d x a d x b , (3.8)the five dimensional scalar Laplacian takes the following form (cid:3) ∆ (cid:16) X α , ˜ X α (cid:17) = 1 √− G ∂ α (cid:16) √− Gg αβ ∂ β ∆ (cid:16) X α , ˜ X α (cid:17)(cid:17) = (cid:18) ∂ ξ + 4 a (cid:48) a ∂ ξ + ∂ c a (cid:19) ∆ (cid:16) X α , ˜ X α (cid:17) . (3.9)Let us now perform the Fourier transform of the Green function ∆ in the transversedirections ( x c ), ∆ (cid:16) X α , ˜ X α (cid:17) = (cid:90) d p (2 π ) e ip ( x − ˜ x ) ∆ p ( ξ, ˜ ξ ) . (3.10)The Fourier components then satisfy (cid:18) ∂ ξ + 4 a (cid:48) a ∂ ξ − p a (cid:19) ∆ p ( ξ, ˜ ξ ) = δ ( ξ − ˜ ξ ) a . (3.11)With a change of variables ˆ∆ p := a ∆ p , this becomes ∂ ξ − p a − (cid:18) a (cid:48) a + a (cid:48)(cid:48) a (cid:19)(cid:124) (cid:123)(cid:122) (cid:125) =4 k ˆ∆ p ( ξ, ˜ ξ ) = 1 a δ ( ξ − ˜ ξ ) , (3.12)The left hand side of (3.12) is exactly the same as that of (3.6). Identifying ˜Σ ab appearingin the right hand side of (3.6) as the source localized on the shell, one can, analogously,write down the scalar propagator for the traceless modes, ˜ χ ab ( p, ξ ). (cid:18) − p a + ∂ ξ − k (cid:19) ∆ ˜ χ ( p ; a + , a − ) = δ ( a + − a − ) . (3.13)This scalar propagator carries most of the relevant features of the bulk graviton propagatorin presence of a localized source on the shell. From now on, we can therefore safely treat themodes, χ ab as scalar modes ˜ χ . The index structure of the actual graviton propagator canbe reinstated using the exact relation between graviton and scalar propagators for localizedsources [20, 25]. The propagator for this mode satisfies equation (3.13) where the location– 11 –f the shell is at a + = a − ≡ a s . Outside the source, this is solved piece-wise inside andoutside the shell by∆ +˜ χ ( p ; a + , a − ) = A ( p, a − ) K (cid:18) pa + k + (cid:19) + B ( p, a − ) I (cid:18) pa + k + (cid:19) , ∆ − ˜ χ ( p ; a + , a − ) = C ( p, a + ) K (cid:18) pa − k − (cid:19) , (3.14)where K and I are modified Bessel functions. I ( p/ak ) diverges at small a , i.e. , at thePoincar´e horizon ( a → − ˜ χ . Continuity of∆ ˜ χ across the shell gives the first junction condition and the stress tensor on the shelldetermines the jump in the derivative. These are the thin shell junction conditions∆ − ˜ χ ( p ; a + , a s ) = ∆ +˜ χ ( p ; a s , a − )12 κ (cid:20) ∂∂ξ i ∆ i ˜ χ ( p ; a + , a − ) (cid:12)(cid:12) a i → a s (cid:21) i =+ i = − + σ +˜ χ ( p ; a s , a − ) = 12 κ . (3.15)Eliminating A ( p, a − ) and C ( p, a + ) from equation (3.13) by imposing these junction condi-tions gives∆ +˜ χ ( p ; a + , a − ) = A ( p, a − ) K (cid:18) pa + k + (cid:19) + B ( p, a − ) I (cid:18) pa + k + (cid:19) = − (cid:20) g K ( p, a s ) + B ( p, a − ) g I ( p, a s ) g K ( p, a s ) (cid:21) K (cid:18) pa + k + (cid:19) + B ( p, a − ) I (cid:18) pa + k + (cid:19) , (3.16)where the functions, g K ( p, a s ) and g I ( p, a s ) are defined as g K ( p, a s ) := pa s K (cid:16) pk − a s (cid:17) (cid:20) K (cid:18) pk − a s (cid:19) K (cid:18) pk + a s (cid:19) − K (cid:18) pk + a s (cid:19) K (cid:18) pk − a s (cid:19)(cid:21) ,g I ( p, a s ) := pa s K (cid:16) pk − a s (cid:17) (cid:20) K (cid:18) pk − a s (cid:19) I (cid:18) pk + a s (cid:19) + I (cid:18) pk + a s (cid:19) K (cid:18) pk − a s (cid:19)(cid:21) . (3.17)Let us examine the behavior of the modes ˜ χ and their propagator near the boundary i.e. , a → ∞ . In this limit one can expand the Bessel functions appearing in the bulk propagator,(3.16)lim a →∞ K (cid:16) pak (cid:17) = 2 a k p −
12 + 4 p log( a ) − p log (cid:0) p k (cid:1) − γp + 3 p a k + . . . , lim a →∞ I (cid:16) pak (cid:17) = 18 p a k + . . . , (3.18)where γ is the Euler-Mascheroni constant. Terms which vanish at the boundary arenormalizable modes in the bulk and yield energy in the holographically dual theory onthe boundary. On the other hand, divergent terms at the boundary are non-normalizablemodes in the bulk and can be interpreted holographically as a change in the boundary– 12 –heory. Since we do not expect a change of energy in the boundary theory in a nucleationevent such as the creation of the brane, we need the normalizable modes to vanish. Formassless scalar modes (also for the traceless modes for graviton fluctuations) in AdS , thiscorresponds to vanishing of terms that go as a − . This fixes the constant, B ( p, a − ) as B ( p, a − ) = − ηηg I ( p, a s ) + g K ( p, a s ) , where η := 3 − γ + 4 ln 2 . (3.19)With this value of B ( p, a − ), the bulk propagator ∆ +˜ χ ( p ; a + , a − ) in equation (3.16), can beevaluated on the shell in the small momentum limit to give∆ +˜ χ ( p ; a + , a − ) = a p (cid:18) k − k + k − − k + (cid:19) + O (cid:0) p (cid:1) ⇒ ∆ shell˜ χ ( p ; a s , a s ) = a s p (cid:18) k − k + k − − k + (cid:19) + O (cid:0) p (cid:1) . (3.20)The leading order term is the four dimensional scalar propagator on the shell. The fivedimensional bulk-bulk propagator in the Randall-Sundrum braneworld [7, 8], as computed in[20], splits into a four dimensional propagator and a piece corresponding to the Kaluza-Kleinmodes. In the above result, we have expanded in small momentum and have recovered thezero mode of the five dimensional graviton corresponding to the four dimensional gravitonlocalized to the shell. The other modes would appear as higher order corrections in p .The expression for the graviton mode on the shell corresponding to equation (3.20),can be obtained by convoluting the propagator with the source, − κ ˜Σ, given in (3.6). Inmomentum space this is a simple product yielding˜ χ s ( p ; a s ) = − κ ˜Σ a s p (cid:18) k − k + k − − k + (cid:19) . (3.21)This is the result that was derived in [5] demonstrating the appearance of four dimensionalgravity with an effective Newton’s constant of G := 2 G k + k − / ( k − − k + ). It is worthmentioning at this point that in this derivation we have worked in a gauge where the braneis placed at ξ = 0. However, as we discussed in section 2, the presence of matter causes thebrane to bend and we must change coordinates accordingly to straighten out the brane. Wesaw that this implies that matter placed directly on the brane, or caused by a string pullingfrom inside, contributes with a negative stress tensor. This implies that ˜Σ ab < T = 3 κ (cid:18) k + − k − (cid:19) p ˜ F , and ˜ h ab = ˜ χ ab + 2 ˜ F η ab + 2 ip [ a ˜ q b ] . (3.22) The log term in (3.18) is a consequence of working with odd-dimensional AdS spacetime. From theholographic renormalization, such terms can be shown to be directly related to the Weyl anomaly of thedual boundary field theory [26]. This log term term can be ignored so long as the boundary metric is flat asis in our case. – 13 –sing equation (3.21) we get˜ h ab = ˜ χ ab + 2 ˜ F η ab + 2 ip [ a ˜ q b ] = − ˜Σ ab αp + 2 ˜ F η ab + 2 ip [ a ˜ q b ] = − αp (cid:18)(cid:18) ˜ T ab −
13 ˜
T η ab (cid:19) + (cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24)(cid:24) κ (cid:18) k + − k − (cid:19) p a p b ˜ F (cid:19) + 2 ˜ F η ab + (cid:24)(cid:24)(cid:24)(cid:24) ip [ a ˜ q b ] , (3.23)where α := 14 κ (cid:18) k + − k − (cid:19) , (3.24)and the arbitrary function ˜ q a is chosen so that is cancels the term proportional to p a p b ˜ F in˜Σ ab ( p ). Using (3.22) this gives˜ h ab = − αp (cid:32) ˜ T ab − ˜ T η ab (cid:33) + ˜ T αp η ab = − αp (cid:32) ˜ T ab − ˜ T η ab (cid:33) = − κ p (cid:32) ˜ T ab − ˜ T η ab (cid:33) , (3.25)where κ ≡ πG . This is precisely the four dimensional junction condition with the rightfour dimensional Planck constant. The minus sign is because of the sign of the stress tensoras discussed before.Let us conclude with few more comments on the main result of this section, equation(3.21). For small momentum, ˜ χ s gives the four dimensional graviton propagator on the shellthereby producing four dimensional gravity. For large momentum, ˜ χ s decays exponentiallyfast towards zero. It can be verified from equations (3.16) and (3.17) that the crossover inthe behavior of ˜ χ s occurs when p ∼ a s k + . This is compatible with a holographic picturewhere the dual CFT on a shell at large and fixed ξ is modified only at large distances whenthe bulk geodesics corresponding to correlation functions in the CFT dip down far enoughinto the bulk to hit the bubble wall. Short distance correlators in the boundary CFT donot see the shell and remain unmodified. So far, we have solved for the traceless perturbation in momentum space. In order to findall components of the metric perturbations in a usable form, we would like to find thecorresponding perturbation in position space. To do this, let us study perturbations of theAdS metric in response to matter on the brane which, as discussed in the previous section,can be the endpoint of a string in the bulk. These perturbations ( χ ab ) can be chosen to beGauss normal and in the Lorenz gauge. Considering spherically symmetric perturbationson the brane , the metric (changing coordinates from ξ to z := e ξ and setting k = 1) can contrary to the perturbation considered in (3.1). – 14 –e written asd s = d z z + z η ab d x a d x b + perturbation (cid:122) (cid:125)(cid:124) (cid:123) χ ab d x a d x b = d z z + z (cid:2) − (1 + h t ( r, z )) d t + (1 + h r ( r, z )) d r + (1 + h a ( r, z )) r dΩ (cid:3) . (3.26)Further, we make an ansatz h i ( r, z ) = z n − f i ( rz ), where rz is the proper radius. Makinga gauge choice that the perturbation is traceless ( h ab η ab = 0) gives the following linearizedEinstein’s equation for f t ( rz ) (cid:0) k z r (cid:1) f (cid:48)(cid:48) t ( zr ) + (cid:18) zr + (2 n + 1) k zr (cid:19) f (cid:48) t ( zr ) + ( n − k f t ( zr ) = 0 , (3.27)where as usual, primes denote derivatives with respect to the argument. In terms of f t , theradial part f r is given by a remarkably simple first order equation zrf (cid:48) r ( zr ) + 3 f r ( zr ) + f t ( zr ) = 0 , (3.28)and the angular part follows from vanishing of the trace f t ( zr ) + f r ( zr ) + 2 f a ( zr ) = 0 . (3.29)For n = 3, equation (3.27) has two solutions f (1) t ( zr ) = 1 zr , f (2) t ( zr ) = 3 + 2 z r (1 + z r ) / . (3.30)The first solution f (1) t ( zr ) gives a perturbation χ tt = z h t = z f t = z /r , while the secondsolution gives a perturbation which has the following asymptoticslim r (cid:28) χ (2) tt ∼ z , lim r (cid:29) χ (2) tt ∼ z r . (3.31)The perturbation χ (1) tt behaves, in fact, like the CHR black string [27] solution at shortdistances and is singular at r = 0. So it needs the presence of string with a delta functionsource δ ( r ). This is a string with non-constant tension [28] and is not related to the constanttension string we study. We discard this solution and focus on χ (2) tt . This gives the followingradial and angular perturbations f r = − √ r z , f a = − r z r z ) / . (3.32)So, the complete perturbed metric to first order isd s = − z (cid:32) c z z r (1 + z r ) / (cid:33) d t + z (cid:18) − c z √ r z (cid:19) d r + z r (cid:32) − c z r z r z ) / (cid:33) dΩ + d z z . (3.33) Choosing h i ( r, z ) = z n − f i ( rz ) makes the perturbation to the AdS metric χ ab ∼ z n f i ( rz ) after includingthe z factor in front of the four dimensional part of the metric. – 15 –his behaves like the CHR solution at large r but does not need a source. The leadingcontribution at large distance is what we used in (2.3) to obtain the Schwarzschild metric onthe brane, (2.4). Taking the brane bending into account, we find that there is an effectivesource, Σ ab = δ ( r ) / κ , induced on the brane as discussed at the end of section 2. Weconclude that the n = 3 solution corresponds to perturbations of the metric due to a pointsource in the brane world (possibly induced by a pulling string) at r = 0. These growtowards the boundary of AdS and are therefore non-normalizable perturbations.Having understood the metric sourced by a mass on the brane, let us now studythe metric induced by the stretched string. In the presence of a source in the bulk, theperturbations can be taken to be cylindrically symmetric around the string but do not needto be in the Lorenz gauge. They are of the general form of equation (3.26) with a non-zeroperturbation in the z direction. Making the ansatz h i ( r, z ) = z n − f i ( rz ) as before, but nowwith n = 0 for a string with constant tension, Einstein’s equations can be solved to linearorder to get the metric outside a constant tension string stretching in the bulk.d s = − z (cid:20) − c z (cid:18) r z − rz (cid:0) r z (cid:1) / (cid:19)(cid:21) d t + z (cid:104) − c z (cid:16) r z − rz (cid:112) r z (cid:17)(cid:105) d r + z r (cid:20) − c z (cid:18) r z − rz (cid:0) r z (cid:1) / (cid:19)(cid:21) dΩ + (cid:20) − c (cid:18) rz (cid:112) r z (cid:19)(cid:21) d z z . (3.34)Away from the string (at large r ), the perturbations go as 1 /z r , while close to thesource (at small r ), h t ∼ h z ∼ /rz . As promised at the beginning of this section, we will conclude with a few comments onhow the computations in the last two sections are connected through a choice of gauge.We showed that depending on whether we choose the brane to be straight or bent, thephysical interpretation would be quite different. In particular, we noted that the bentgauge automatically allows for an effective delta function source on the brane. This iswhat connects our computation in momentum space to that in the position space. Whilein the first case, we kept the position of the brane fixed, in the later we allowed branebending. Noting that the Fourier transform of the solution f (2) t ( zr ) in (3.30) is nothing butthe modified Bessel function K (see appendix A), we intuitively see that if we had chosento work out the fluctuations in the momentum space, but taking into account the bendingof the brane, we would have needed to work with the pure K solution on both sides ofthe brane. This would amount to setting B ( p, a − ) = 0 in (3.14). By self consistency, then,the brane is bent so that the presence of the K does not induce any matter on the brane,such that we find a pure Schwarzschild metric. Alternatively, we can perform a coordinatetransformation, shifting z as a function of the coordinates on the brane, to obtain a straightbrane. This will generate terms of the form I through mixing. Again, we should make– 16 –ure that there is no matter induced on the brane, which corresponds to there being nonormalizable modes. This implies a specific relative coefficient of the K and I so thatthe subleading 1 /z mode is canceled. This would precisely correspond to the momentumspace computation that we have performed. In this work, we have provided a clear understanding of backreaction on our shellworlddue to localized and extended sources in the bulk. This complete picture, in turn, suggestsseveral interesting directions for further studies that we are currently pursuing.
We have argued that there is a consistent holographic picture of the dark brane, withmassive particles realized as strings hanging from the boundary of AdS. Even thoughgravity is mediated by the bending of the brane, gravitational effects extend all the way tothe boundary through the non-normalizable modes. Interestingly, there is an alternativepossibility involving two dark branes where gravity is fully localized in the fifth direction.Physically, it would correspond to two branes about to collide. As shown in figure 2, onecan imagine strings stretching between the two branes, making each of the branes bend inresponse. Our universe, together with the matter it contains, is supported on this combinedstructure. Thus, it is perhaps better to think of the two branes as one thick brane with aninternal structure. As the universe expands, the sandwich gets thinner, implying a dramaticend point when the bubbles collide. As explained in [6], the distance between the twobranes can be microscopic and still allow for plenty of proper time on the branes beforecollision.In this picture, any gravitational perturbation is localized to the sandwich universeand decays as we move away from it. The holographic picture corresponds to slicing thesandwich in two, assuming the two sides to be mirrors of each other, and inserting aholographic screen. It also requires that the two branes are sufficiently far away from eachother. We will investigate this setup further in a forthcoming paper.
An important challenge is to construct solutions corresponding to black holes. Given thatmassive particles corresponds to stretched strings, it seems natural to consider such objectsin the context of the sandwich universe discussed above. Black holes should form if, forinstance, a shell of stretched strings such as in figure 2, were to collide and approach theirSchwarzschild radius. One possible outcome would be a five dimensional black hole actingas a connection between the two braneworlds. Inspired by [29, 30], we believe there arealso other possibilities that should be investigated. In particular, the collision of the stringscould initiate the nucleation of a bubble of true vacuum that could prevent the formationof a black hole. The idea would be that such a structure could remain stable due to Unruhradiation in parallel to what was proposed in [29, 30]. We will provide details of suchconstructions in an upcoming publication.– 17 – a ξ k − k + k + k − k + Figure 2 : A cartoon representing strings (in red) stretching between two copies of theworld brane (in black) that can be close together. The energy density of the strings andtheir endpoints on the branes causes them to bend. As the four dimensional universe withinthe brane expands, the branes move closer to each other (shown with blue arrows). Thegravitational perturbation from the strings is localized to the branes and the false vacuum inbetween. The perturbations decays away as we move out from the sandwich universe intothe true vacuum.
The derivation of the bulk propagator in momentum space presented in section 3.1 showsthat the effective four-dimensional gravity on our shellworld is perfectly consistent with theexpectations from holography. By holography we here mean the description of gravitationalphysics in the bulk AdS in terms of the field theory living on its boundary at a = ∞ . Here,it is worth mentioning several previous attempts towards investigating the gravitationalduals of collapsing shells in asymptotically AdS spacetime [31, 32, 33, 34]. However, with aview to understand the gravitational dual of the dynamics of thermalization in the boundaryquantum field theory, all of these studies were centered on collapsing shells. It wouldnevertheless be straightforward to do a similar computation of boundary propagators inour case. Although our model of expanding bubble is physically very different from thescenario of collapsing shells in AdS, the holographic techniques we need to employ in orderto compute the boundary correlators would be quite similar. In particular, taking thebackreaction into account, it would be worth investigating the signatures of a possibleformation on black objects on the shell from the perspective of the boundary field theory[35]. However, it would perhaps be even more interesting and illuminating to understandthe whole picture in terms of the the holography on the shell. Considering the shell to be afinite radial cut off from the perspective of both the AdS spacetimes, one possible way to– 18 –nderstand this holographically is to consider a CFT deformed by an irrelevant operator,typically termed as T ¯ T deformation in literature [36, 37, 38]. This is a very active fieldof research in the holography community and would be instrumental in connecting thefive-dimensional bulk physics in our set up to a deformed CFT on the shellworld. We hopeto report on this soon. Acknowledgments
We would like thank Giuseppe Dibitetto, Johanna Erdmenger, Miguel Montero, Lisa Randall,Marjorie Schillo, Cumrun Vafa and Irene Valenzuela for several stimulating discussions.We would also like to thank the referees for their valuable comments. The work of S.B. issupported by the Alexander von Humboldt postdoctoral fellowship.
A Fourier transform of K The 3 dimensional Fourier transform of the modified Bessel function K in sphericalcoordinates is given by F ( K ( p )) = (cid:90) d (cid:126)pe i(cid:126)p · (cid:126)r K ( p ) = 2 π π (cid:90) d θ sin θ (cid:90) d p e ipr cos θ p K ( p )= 4 πr (cid:90) d p p sin prK ( p )= − πr ∂∂r (cid:90) d p cos prK ( p ) (A.1) K ( p ) has an integral representation given by [39] K ( p ) = − ∞ (cid:90) −∞ d r cos pr r + 12 √ r , (A.2)which can be inverted to give ∞ (cid:90) −∞ d p cos prK ( p ) = − π r + 1 √ r , (A.3)Using this in (A.1) gives F ( K ( p )) = 4 πr ∂∂r (cid:18) r + 12 π √ r (cid:19) = 4 π r + 3(1 + r ) / . (A.4) The Fourier cosine transform and its inverse are defined with normalization F ( ω ) = (1 / ∞ (cid:82) −∞ f ( t ) cos ωt d t and f ( t ) = (1 /π ) ∞ (cid:82) −∞ F ( ω ) cos ωt d ω respectively. – 19 – eferences [1] S. K. Garg and C. Krishnan, Bounds on Slow Roll and the de Sitter Swampland , JHEP (2019) 075 [ ].[2] G. Obied, H. Ooguri, L. Spodyneiko and C. Vafa, De Sitter Space and the Swampland , .[3] H. Ooguri, E. Palti, G. Shiu and C. Vafa, Distance and de Sitter Conjectures on theSwampland , Phys. Lett.
B788 (2019) 180–184 [ ].[4] U. H. Danielsson and T. Van Riet,
What if string theory has no de Sitter vacua? , Int. J. Mod.Phys.
D27 (2018), no. 12, 1830007 [ ].[5] S. Banerjee, U. Danielsson, G. Dibitetto, S. Giri and M. Schillo,
Emergent de Sitter Cosmologyfrom Decaying Anti–de Sitter Space , Phys. Rev. Lett. (2018), no. 26, 261301 [ ].[6] S. Banerjee, U. Danielsson, G. Dibitetto, S. Giri and M. Schillo, de Sitter Cosmology on anexpanding bubble , JHEP (2019) 164 [ ].[7] L. Randall and R. Sundrum, A Large mass hierarchy from a small extra dimension , Phys. Rev.Lett. (1999) 3370–3373 [ hep-ph/9905221 ].[8] L. Randall and R. Sundrum, An Alternative to compactification , Phys. Rev. Lett. (1999)4690–4693 [ hep-th/9906064 ].[9] I. Koga and Y. Ookouchi, Catalytic Creation of Baby Bubble Universe with Small PositiveCosmological Constant , JHEP (2019) 281 [ ].[10] H. Ooguri and C. Vafa, Non-supersymmetric AdS and the Swampland , Adv. Theor. Math.Phys. (2017) 1787–1801 [ ].[11] U. Danielsson and G. Dibitetto, Fate of stringy AdS vacua and the weak gravity conjecture ,Phys. Rev.
D96 (2017), no. 2, 026020 [ ].[12] U. H. Danielsson, G. Dibitetto and S. C. Vargas,
A swamp of non-SUSY vacua , JHEP (2017) 152 [ ].[13] N. Arkani-Hamed, L. Motl, A. Nicolis and C. Vafa, The String landscape, black holes andgravity as the weakest force , JHEP (2007) 060 [ hep-th/0601001 ].[14] L. Susskind, The World as a hologram , J. Math. Phys. (1995) 6377–6396 [ hep-th/9409089 ].[15] J. M. Maldacena, The Large N limit of superconformal field theories and supergravity , Int. J.Theor. Phys. (1999) 1113–1133 [ hep-th/9711200 ], [Adv. Theor. Math. Phys.2,231(1998)].[16] E. Witten, Anti-de Sitter space and holography , Adv. Theor. Math. Phys. (1998) 253–291[ hep-th/9802150 ].[17] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Gauge theory correlators from noncriticalstring theory , Phys. Lett.
B428 (1998) 105–114 [ hep-th/9802109 ].[18] O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri and Y. Oz,
Large N field theories,string theory and gravity , Phys. Rept. (2000) 183–386 [ hep-th/9905111 ]. – 20 –
19] J. Garriga and T. Tanaka,
Gravity in the brane world , Phys. Rev. Lett. (2000) 2778–2781[ hep-th/9911055 ].[20] S. B. Giddings, E. Katz and L. Randall, Linearized gravity in brane backgrounds , JHEP (2000) 023 [ hep-th/0002091 ].[21] W. Israel, Singular hypersurfaces and thin shells in general relativity , Nuovo Cim.
B44S10 (1966) 1 [Erratum: Nuovo Cim.B48,463(1967); Nuovo Cim.B44,1(1966)].[22] K. Lanczos,
Fl¨achenhafte Verteilung der Materie in der Einsteinschen Gravitationstheorie ,Annalen der Physik (1924), no. 14, 518–540[ https://onlinelibrary.wiley.com/doi/pdf/10.1002/andp.19243791403 ].[23] N. Sen, ¨Uber die Grenzbedingungen des Schwerefeldes an Unstetigkeitsfl¨achen , Annalen derPhysik (1924), no. 5-6, 365–396[ https://onlinelibrary.wiley.com/doi/pdf/10.1002/andp.19243780505 ].[24] A. Padilla,
Infra-red modification of gravity from asymmetric branes , Class. Quant. Grav. (2005), no. 6, 1087–1104 [ hep-th/0410033 ].[25] E. D’Hoker, D. Z. Freedman, S. D. Mathur, A. Matusis and L. Rastelli, Graviton and gaugeboson propagators in AdS(d+1) , Nucl. Phys.
B562 (1999) 330–352 [ hep-th/9902042 ].[26] S. de Haro, S. N. Solodukhin and K. Skenderis,
Holographic reconstruction of space-time andrenormalization in the AdS / CFT correspondence , Commun. Math. Phys. (2001) 595–622[ hep-th/0002230 ].[27] A. Chamblin, S. W. Hawking and H. S. Reall,
Brane world black holes , Phys. Rev.
D61 (2000)065007 [ hep-th/9909205 ].[28] R. B. Mann, E. Radu and C. Stelea,
Black string solutions with negative cosmological constant ,JHEP (2006) 073 [ hep-th/0604205 ].[29] U. H. Danielsson, G. Dibitetto and S. Giri, Black holes as bubbles of AdS , JHEP (2017) 171[ ].[30] U. Danielsson and S. Giri, Observational signatures from horizonless black shells imitatingrotating black holes , JHEP (2018) 070 [ ].[31] U. H. Danielsson, E. Keski-Vakkuri and M. Kruczenski, Vacua, propagators, and holographicprobes in AdS / CFT , JHEP (1999) 002 [ hep-th/9812007 ].[32] U. H. Danielsson, E. Keski-Vakkuri and M. Kruczenski, Spherically collapsing matter in AdS,holography, and shellons , Nucl. Phys.
B563 (1999) 279–292 [ hep-th/9905227 ].[33] U. H. Danielsson, E. Keski-Vakkuri and M. Kruczenski,
Black hole formation in AdS andthermalization on the boundary , JHEP (2000) 039 [ hep-th/9912209 ].[34] S. B. Giddings and A. Nudelman, Gravitational collapse and its boundary description in AdS ,JHEP (2002) 003 [ hep-th/0112099 ].[35] R. Gregory, S. F. Ross and R. Zegers, Classical and quantum gravity of brane black holes ,JHEP (2008) 029 [ ].[36] L. McGough, M. Mezei and H. Verlinde, Moving the CFT into the bulk with
T T , JHEP (2018) 010 [ ]. – 21 –
37] M. Taylor,
TT deformations in general dimensions , .[38] T. Hartman, J. Kruthoff, E. Shaghoulian and A. Tajdini, Holography at finite cutoff with a T deformation , JHEP (2019) 004 [ ].[39] NIST Digital Library of Mathematical Functions, Equation 10.32.7 . http://dlmf.nist.gov/10.32 , Release 1.0.25 of 2019-12-15. F. W. J. Olver, A. B. OldeDaalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V.Saunders, H. S. Cohl, and M. A. McClain, eds., Release 1.0.25 of 2019-12-15. F. W. J. Olver, A. B. OldeDaalhuis, D. W. Lozier, B. I. Schneider, R. F. Boisvert, C. W. Clark, B. R. Miller, B. V.Saunders, H. S. Cohl, and M. A. McClain, eds.